We turn now to the verification of conditions on the relative contraction rates. Such conditions are typically given in a quite general setting in terms of Lipschitz-type bounds for the components of the map (REF ). Our aim here is to restate these conditions in terms of spectral properties like normal hyperbolicity which, as a spectral condition, is invariant under \(C^1-\) smooth coordinate transformations and thus detectable in the original coordinates of (REF ). The cost of this reformulation is that we require at least \(C^1-\) smoothness of the map, while only Lipschitz continuity is required in [1]}. A similar price is paid in Fenichel theory [2]}, [3]}.
